There are numerous prior art airfoil and hydrofoil structures, such as a common commercial airplane wing. The surface textures of such structures are typically smooth or include small surface protrusions such as pop rivets and the like. All of such surfaces are typically defined by Euclidean geometry and produce well-known turbulence effects.
Many fluid dynamics phenomena, such as aerodynamic turbulence, however, do not possess Euclidean geometric characteristics. They can, on the other hand, be analyzed using fractal geometry. Fractal geometry comprises an alternative set of geometric principles conceived and developed by Benoit B. Mandelbrot. An important treatise on the study of fractal geometry is Mandelbrot's The Fractal Geometry of Nature.
As discussed in Mandelbrot's treatise, many forms in nature are so irregular and fragmented that Euclidean geometry is not adequate to represent them. In his treatise, Mandelbrot identified a family of shapes, which described the irregular and fragmented shapes in nature, and called them fractals. A fractal is defined by its topological dimension DT and its Hausdorf dimension D. As defined, DT is always an integer, D need not be an integer, and D≥DT. (See p. 15 of Mandelbrot's The Fractal Geometry of Nature). Fractals may be represented by two-dimensional shapes and three-dimensional objects. In addition, fractals possess self-similarity in that they have the same shapes or structures on both small and large scales.
It has been found that fractals have characteristics that are significant in a variety of fields. For example, fractals correspond with naturally occurring phenomena such as aerodynamic phenomena. In addition, three-dimensional fractals have very specific electromagnetic wave-propagation properties that lead to special wave-matter interaction modes. Fractal geometry is also useful in describing naturally occurring forms and objects such as a stretch of coastline. Although the distance of the stretch may be measured along a straight line between two points on the coastline, the distance may be more accurately considered infinite as one considers in detail the irregular twists and turns of the coastline.
Fractals can be generated based on their property of self-similarity by means of a recursive algorithm. In addition, fractals can be generated by various initiators and generators as illustrated in Mandelbrot's treatise.
An example of a three-dimensional fractal is illustrated in U.S. Pat. No. 5,355,318 to Dionnet et al., the entire contents of which are incorporated herein by reference. The three-dimensional fractal described in this patent is referred to as Serpienski's mesh. This mesh is created by performing repeated scaling reductions of a parent triangle into daughter triangles until the daughter triangles become infinitely small. The dimension of the fractal is given by the relationship (log N)/(log E) where N is the number of daughter triangles in the fractal and E is a scale factor.
Some processes for making self-similar three-dimensional fractals is known. For example, the Dionnet et al. patent discloses methods of enabling three-dimensional fractals to be manufactured. The method consists in performing repeated scaling reductions on a parent generator defined by means of three-dimensional coordinates, in storing the coordinates of each daughter object obtained by such a scaling reduction, and in repeating the scaling reduction until the dimensions of a daughter object become less than a given threshold value. The coordinates of the daughter objects are then supplied to a stereolithographic apparatus which manufactures the fractal defined by assembling together the daughter objects.
In addition, U.S. Pat. No. 5,132,831 to Shih et al. discloses an analog optical processor for performing affine transformations and constructing three-dimensional fractals that may be used to model natural objects such as trees and mountains. An affine transformation is a mathematical transformation equivalent to a rotation, translation, and contraction (or expansion) with respect to a fixed origin and coordinate system. There are also a number of prior-art patents directed toward two-dimensional fractal image generation. For example, European Patent No. 0 463 766 A2 to Applicant GEC-Marconi Ltd. discloses a method of generating fractal images representing fractal objects. This invention is particularly applicable to the generation of terrain images. In addition, U.S. Pat. No. 4,694,407 to Ogden discloses fractal generation, as for video graphic displays. Two-dimensional fractal images are generated by convolving a basic shape, or “generator pattern,” with a “seed pattern” of dots, in each of different spatial scalings.
Fractal patterns have be used for radio receivers and transceivers, as described in U.S. Pat. No. 6,452,553 to Cohen, and U.S. Pat. No. 7,126,537 to Cohen, the entire contents of both of which are incorporated herein by reference. See also Hohlfeld, R., and Cohen, N., “SELF-SIMILARITY AND THE GEOMETRIC REQUIREMENTS FOR FREQUENCY INDEPENDENCE IN ANTENNAE,” Fractals, Vol. 7, No. 1 (1999) 79-84, the entire contents of which are incorporated herein by reference.
Thus, as current techniques for shaping airfoils, hydrofoils, and other fluid-contact surfaces are based on Euclidean geometries, such surfaces create undesirable turbulences effects, including reduced fuel efficiency and reduced maneuverability. Additional undesirable turbulence effects can include the potentially deleterious eddy currents or vortexes produced by large scale commercial aircraft, which can pose problems or hazards for other aircraft including smaller commercial and private aircraft. Consequently, there a need exists to improve surfaces of airfoils and hydrofoils for reduced drag and improved manuererability characteristics.